Name and significance of this subset of a semiring

Question

Let $ (S, \oplus, \otimes)$ be a commutative semiring and $H$ be a subset of $ S$ such that for all $ x, y\in H $ implies that $x\otimes y= e$ where $e$ is the multiplicative identity( i.e., the identity with respect to the operation $ \otimes$). Then what is the name of such subset of a semiring? In other words, is there any specific name for the subset $H$? Are there any significant properties of such structure of $H$ in ring or semiring theory\application?

Answer 1

This property is rather trivial: it just means that either $H$ is empty or $H$ consists of a single element $x$ such that $x\otimes x=e$. Indeed, if $x,y\in H$, then $x\otimes y=x\otimes x=e$ and so $y=(x\otimes x)\otimes y=x\otimes (x\otimes y)=x$ and so $H$ can have at most one element.